arXiv research

The paper characterizes convex co-compact groups with one-dimensional boundary faces.

New rigidity result for convex co-compact actions in products of spaces.

The main purpose of this note is to provide a topological approach to defining additive functions on Riemannian co-compact normal coverings.

We survey some results and questions about free actions of infinite groups on products of spheres and euclidean spaces, and give some new co-compact examples.

3-manifold groups can only have convex co-compact representations if they are geometric or hyperbolic.

We strengthen the analogy between convex co-compact Kleinian groups and convex co-compact subgroups of the mapping class group of a surface (in the sense of B. Farb and L. Mosher).

3D hyperbolic manifolds map one-to-one to their boundary character varieties.

New manifold types defined on quotient spaces.

Unique hyperbolic manifolds identified by boundary pleating.

The renormalized volume is reinterpreted using isoperimetric profiles.

Study shows infimum of dual volume equals convex core volume for hyperbolic 3-manifolds.

Study on Cheeger constant in specific hyperbolic 3-manifolds.

The study shows how certain surfaces can be filled by hyperbolic manifolds.

For convex co-compact hyperbolic manifolds $Γ\backslash \mathbb{H}^{n+1}$ for which the dimension of the limit set satisfies $δ_Γ< n/2$, we show that the high-frequency Eisenstein series associated to a point $ξ$ "at infinity" concentrate microlocally on a measure supported by (the closure of) the set of points in the …

For quotients of the $n+1$-dimensional hyperbolic space by a convex co-compact group $Γ$, we obtain a formula relating the renormalized trace of the wave operator with the resonances of the Laplacian and some conformal invariants of the boundary, generalizing a formula of Guillopé and Zworski in dimension 2. By writing…

We show that every finitely-generated free subgroup of a right-angled, co-compact Kleinian reflection group is contained in a surface subgroup.

The paper studies the action of automorphisms on train tracks and finds that the set of minimally displaced points is co-compact.

We prove that a compact nilmanifold admits a Sasakian structure if and only if it is a compact quotient of the generalized Heisenberg group of odd dimension by a co-compact discrete subgroup.

For compact and for convex co-compact oriented hyperbolic surfaces, we prove an explicit correspondence between classical Ruelle resonant states and quantum resonant states, except at negative integers where the correspondence involves holomorphic sections of line bundles.

We show meromorphic extension and analyze the divisors of a Selberg zeta function of odd type $Z_{Γ,Σ}^{\rm o}(λ)$ associated to the spinor bundle $Σ$ on odd dimensional convex co-compact hyperbolic manifolds $X:=Γ\backslash\hh^{2n+1}$. We define a natural eta invariant $η(D)$ associated to the Dirac operator $D$ on $X…

New adapted renormalized volume for hyperbolic 3-manifolds with compressible boundary.

We prove a dynamical wave trace formula for asymptotically hyperbolic (n+1) dimensional manifolds with negative (but not necessarily constant) sectional curvatures which equates the renormalized wave trace to the lengths of closed geodesics. A corollary of this dynamical trace formula is a dynamical resonance-wave trac…

In this paper, we consider discrete groups in ${\rm PGL}_d(\mathbb{R})$ acting convex co-compactly on a properly convex domain in real projective space. For such groups, we establish an analogue of the well known flat torus theorem for ${\rm CAT}(0)$ spaces.

Study on hyperbolic manifolds and their boundary data, focusing on volume functions.

We show that a compactly generated locally compact group of polynomial growth having no non-trivial compact normal subgroups can be embedded as a co-compact subgroup into a semidirect product of a connected, simply connected, nilpotent Lie group and a compact group. There is also a uniqueness statement for this extensi…

For convex co-compact hyperbolic quotients $X=Γ\backslash\hh^{n+1}$, we analyze the long-time asymptotic of the solution of the wave equation $u(t)$ with smooth compactly supported initial data $f=(f_0,f_1)$. We show that, if the Hausdorff dimension $δ$ of the limit set is less than $n/2$, then $u(t) = C_δ(f) e^{(δ-\nd…

For every dimension d, there is an infinite family of convex co-compact reflection groups of isometries of hyperbolic d-space --- the superideal (simplicial and cubical) reflection groups --- with the property that a random group at any density less than a half (or in the few relators model) contains quasiconvex subgro…

Flow on curves in inversive geometry converges to loxodromics.

We define and study the renormalized volume for geometrically finite hyperbolic $3$-manifolds, including with rank-$1$ cusps. We prove a variation formula, and show that for certain families of convex co-compact hyperbolic metrics $g_\eps$ degenerating to a geometrically finite hyperbolic metric $g_0$ with rank-$1$ cus…

We provide an equivariant description/classification of all complete (compact or not) non-negatively curved manifolds M together with a co-compact action by a reflection group W, and moreover, classify such W. In particular, we show that the building blocks consist of the classical constant curvature models and general…

We give a new upper bound on the Selberg zeta function for a convex co-compact Schottky group acting on ${\mathbb H}^{n+1}$: in strips parallel to the imaginary axis the zeta function is bounded by $\exp (C |s|^δ)$ where $δ$ is the dimension of the limit set of the group. This bound is more precise than the optimal…

Automorphisms of Riemann surfaces with specific properties are studied.

No exact G₂-structures on compact Lie group quotients.

In this paper we consider discrete groups in ${\rm PGL}_d(\mathbb{R})$ acting convex co-compactly on a properly convex domain in real projective space. For such groups, we establish necessary and sufficient conditions for the group to be relatively hyperbolic in terms of the geometry of the convex domain. This answers …

We extend harmonic map techniques to the setting of more general differential equations in conformal geometry. We obtain an extension of Siu's rigidity to Kahler-Weyl geometry and apply the latter to Vaisman's conjecture. Other applications include topological obstructions to the existence of Kahler-Weyl structures. Fo…

Take two isomorphic convex co-compact co-infinite volume Kleinian groups, whose regular sets are diffeomorphic. The quotient of hyperbolic 3-space by these groups gives two hyperbolic 3-manifolds whose scattering operators may be compared. We prove that the operator norm of the difference between the scattering operato…

We present examples of hyperbolizable 3-manifolds $M$ with the following property. Let $CC(π_1(M))$ denote the space of convex co-compact representations of $π_1(M)$. We show that for every $K\geq 1$ there exists a representation $ρ$ in $\bar {CC(π_1(M))}$ so that every $K$-quasiconformal deformation of $ρ$ lies in the…

The main results of this article provide asymptotics at infinity of the Green's functions near and at the spectral gap edges for "generic" periodic second-order elliptic operators on noncompact Riemannian co-compact coverings with abelian deck groups. Previously, analogous results have been known for the case of $\math…

We prove a criterion for the existence of harmonic metrics on Higgs bundles that are defined on smooth loci of klt varieties. As one application, we resolve the quasi-etale uniformisation problem for minimal varieties of general type to obtain a complete numerical characterisation of singular quotients of the unit ball…

In this paper we investigate the higher dimensional divergence functions of mapping class groups of surfaces and of CAT(0)--groups. We show that, for mapping class groups of surfaces, these functions exhibit phase transitions at the rank (as measured by thrice the genus plus the number of punctures minus 3). We also pr…

In this paper, it is shown that every point in the hyperbolic 3-space is moved at a distance at least $0.5\log\left(12\cdot 3^{k-1}-3\right)$ by one of the isometries of length at most $k\geq 2$ in a 2-generator Klenian group $Γ$ which is torsion-free, not co-compact and contains no parabolic. Also some lower bounds fo…

We consider 3-dimensional hyperbolic cone-manifolds, singular along infinite lines, which are ``convex co-compact'' in a natural sense. We prove an infinitesimal rigidity statement when the angle around the singular lines is less than $π$: any first-order deformation changes either one of those angles or the conformal …

For G an almost-connected Lie group, we study G-equivariant index theory for proper co-compact actions with various applications, including obstructions to and existence of G-invariant Riemannian metrics of positive scalar curvature. We prove a rigidity result for almost-complex manifolds, generalising Hattori's result…

Let $Γ$ be a (non-elementary) convex co-compact group of isometries of a pinched Hadamard manifold $X$. We show that a normal subgroup $Γ_0$ has critical exponent equal to the critical exponent of $Γ$ if and only if $Γ/ Γ_0$ is amenable. We prove a similar result for the exponential growth rate of closed geodesics on $…

Formula derived for FUP exponent in quasi-Fuchsian groups.

In this note, we study the dynamics and associated zeta functions of conformally compact manifolds with variable negative sectional curvatures. We begin with a discussion of a larger class of manifolds known as convex co-compact manifolds with variable negative curvature. Applying results from dynamics on these spaces,…

We study integrability of generalized almost contact structures, and find conditions under which the main associated maximal isotropic vector bundles form Lie bialgebroids. These conditions differentiate the concept of generalized contact structures from a counterpart of generalized complex structures on odd-dimensiona…

A nilmanifold is a quotient of a nilpotent group $G$ by a co-compact discrete subgroup. A complex nilmanifold is one which is equipped with a $G$-invariant complex structure. We prove that a complex nilmanifold has trivial canonical bundle. This is used to study hypercomplex nilmanifolds (nilmanifolds with a triple of …